Well, I have a clever way to explain the relationship between areas under a hyperbola and continuously compounded interest: dy/dx = 1/x dx/dy = x Hopefully t

Well, I have a clever way to explain the relationship between areas under a hyperbola and continuously compounded interest: dy/dx = 1/x dx/dy = x Hopefully that isn't too confusing. If you start with the second relation (continuously compounded interest), a reciprocal gets you to the first (area under a hyperbola) -- the connection between exponentials/logarithms and areas under hyperbolae isn't immediately obvious to most people. Implicit differentiation is kind of scary, but "draw a curve whose slope is its height" requires an art degree. I like your essay on intuition, but I don't know if your interpretation of the factorial series is really deep enough to explain what is going on. A fully elementary treatment of Taylor's series is a really hard thing to achieve. This, however, could help: http://en.wikipedia.org/wiki/Binomial_theorem#Series_for_e http://en.wikipedia.org/wiki/Binomial_theorem#Series_for_e The binomial theorem for integer exponents is easy enough to grasp, though Wikipedia's geometrical explanation is kind of hilarious: "[...] if one sets a = x and b = Δx, interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, [...]" I'm afraid infintesimal changes in the volume of n-dimensional hypercubes probably aren't going to make exponential functions any easier to grasp for epsilons...